Closed graph property in Alexandroff spaces

TL;DR

This paper characterizes when functions from Alexandroff spaces have closed graphs, showing they are constant on connected components and continuous, with the number of such maps depending on the space's connected components and closed points.

Contribution

It provides a complete characterization of functions with closed graphs from Alexandroff spaces, linking their properties to connected components and closed points.

Findings

Functions with closed graphs are constant on connected components.

Such functions are necessarily continuous.

Number of these functions depends on connected components and closed points.

Abstract

In the following text we show if $X$ is an Alexandroff space, then $f:X\to Y$ has closed graph if and only if it has constant closed value on each connected component of $X$. Moreover, if $X$ an Alexandroff space and $f:X\to Y$ has closed graph, then $f:X\to Y$ is continuous. As a matter of fact, the number of maps which have closed graph from Alexandroff space $X$ to a topological space $Y$ depends just on the the number of connected components of $X$ and the number of closed points of $Y$.